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instances - Maxwell's and Einstein's - are of a different order
from the first: in both, totc/Hy unknown and unforeseen pheno-
mena were predicted mathematically; that is, these predictions
were qualitative. Both Maxwell and Einstein amplified their
qualitative foresight by precise quantit'itive predictions which
precluded any charge of mere guessing when Iheir prophecies
were finally verified experimentally.
Hamilton's prediction of what is called conical refraction in
optics was of this same qualitative plus quantitative order.
From his theory of systems of rays he predicted mathematically
that a wholly unexpected phenomenon would be found in con-
nexion with the refraction of light in biaxial crystals. While
polishing the Third Supplement to his memoir on rays he sur-
prised himself by a discovery which he thus describes:
'The law of the reflection of light at ordinary mirrors appears
to have been known to Euclid; that of ordinary refraction at a
surface of water, glass, or other uncrystallized medium, was
discovered at a much later date by SneUius; Huygens disco-
vered, and Malus confirmed, the law of extraordinary refraction
produced by uniaxal crystals, such as Iceland spar; and finally,
the law of the extraordinary double refraction at the faces of
biaxal crystals, such as topaz or arragonite, was found hi our
own tune by Fresnel. But even in these cases of extraordinary
or crystalline refraction, no more than two refracted rays had
ever been observed or even suspected to exist, if we except a
theory of Cauchy, that there might possibly be a third ray,
though probably imperceptible to our senses. Professor Hamil-
ton, however, in investigating by his general method the conse-
quences of the law of Fresnel, was led to conclude that there
ought to be hi certain cases, which he assigned* not merely two,
nor three, nor any finite number, but an infinite number, or a
cone of refracted rays within a biaxal crystal, corresponding to
and resulting from a single incident ray; and that in certain
other cases, a single ray within such a crystal should give rise
to an infinite number of emergent rays, arranged in a certain
other cone. He was led, therefore, to anticipate from theory
two new laws of light, to which he gave the names of Internal
and External Conical Refraction.*